Question

a fraction becomes 9 by 11 is 2 is added to both the numerator and the denominator if3 is added to both the numerator and the denominator it becomes 5by6 find the fraction ​

Answer 1

Answer:

Let the fraction is x/y. Then by the given question,

(x+2)/(y+2)=9/11

or, 11x+22=9y+18

or, 11x-9y=18-22

or, 11x-9y=-4

or, 11x=9y-4

or, x=(9y-4)/11

and, (x+3)/(y+3)=5/6

or, 6x+18=5y+15

or, 6x-5y=15-18

or, 6(9y-4)/11-5y=-3

or, (54y-24-55y)11=-3

or, -y-24=-3×11

or, y+24=33

or, y=33-24=9

then, x=(9×9-4)/11

or, x=(81-4)/11

or, x=77/11=7

∴, x=7,y=9 and the fraction is :7/9

Answer 2

Let numerator be "N" and denominator be "D".

a fraction becomes 9/11 is 2 is added to both the numerator and the denominator.

According to question,

$\Rightarrow\:\frac{N\:+\:2}{D\:+\:2}\:=\:\frac{9}{11}$

Cross multiply them

$\Rightarrow\:11(N\:+\:2)\:=\:9(D\:+\:2)$

$\Rightarrow\:11N\:+\:22\:=\:9D\:+\:18$

$\Rightarrow\:11N\:=\:9D\:+\:18\:-\:22$

$\Rightarrow\:11N\:=\:9D\:-\:4$ ____ (eq 1)

If 3 is added to both the numerator and the denominator it becomes 5/6.

$\Rightarrow\:\frac{N\:+\:3}{D\:+\:3}\:=\:\frac{5}{6}$

Cross multiply them

$\Rightarrow\:6(N\:+\:3)\:=\:5(D\:+\:3)$

$\Rightarrow\:6N\:+\:18\:=\:5D\:+\:15$

$\Rightarrow\:6N\:=\:5D\:+\:15\:-\:18$

$\Rightarrow\:6N\:=\:5D\:-\:3$ ____ (eq 2)

On multiplying (eq 1) with 6 and (eq 2) with 11 we get,

$\Rightarrow\:66N\:=\:54D\:-\:24$ ____ (eq 3)

$\Rightarrow\:66N\:=\:55D\:-\:33$ ____ (eq 4)

On comparing (eq 3) and (eq 4) we get,

$\Rightarrow\:54D\:-\:24\:=\:55D\:-\:33$

$\Rightarrow\:55D\:-\:54D\:=\:33\:-\:24$

$\Rightarrow\:D\:=\:9$

Put value of D in (eq 1)

$\Rightarrow\:11N\:=\:9(9)\:-\:4$

$\Rightarrow\:11N\:=\:81\:-\:4$

$\Rightarrow\:11N\:=\:77$

$\Rightarrow\:N\:=\:7$

∴ $\bold{Fraction\:=\:\dfrac{7}{9}}$